# A Unified Approach to the Klein–Gordon Equation on Bianchi Backgrounds

## Abstract

In this paper, we study solutions to the Klein–Gordon equation on Bianchi backgrounds. In particular, we are interested in the asymptotic behaviour of solutions in the direction of silent singularities. The main conclusion is that, for a given solution *u* to the Klein–Gordon equation, there are smooth functions *u*_{i}, *i* = 0,1, on the Lie group under consideration, such that \({u_{\sigma}(\cdot,\sigma)-u_{1}}\) and \({u(\cdot,\sigma)-u_{1}
\sigma-u_{0}}\) asymptotically converge to zero in the direction of the singularity (where \({\sigma}\) is a geometrically defined time coordinate such that the singularity corresponds to \({\sigma\rightarrow-\infty}\)). Here *u*_{i}, *i* = 0, 1, should be thought of as data on the singularity. Interestingly, it is possible to prove that the asymptotics are of this form for a large class of Bianchi spacetimes. Moreover, the conclusion applies for singularities that are matter dominated; singularities that are vacuum dominated; and even when the asymptotics of the underlying Bianchi spacetime are oscillatory. To summarise, there seems to be a universality as far as the asymptotics in the direction of silent singularities are concerned. In fact, it is tempting to conjecture that as long as the singularity of the underlying Bianchi spacetime is silent, then the asymptotics of solutions are as described above. In order to contrast the above asymptotics with the non-silent setting, we, by appealing to known results, provide a complete asymptotic characterisation of solutions to the Klein–Gordon equation on a flat Kasner background. In that setting, \({u_{\sigma}}\) does, generically, not converge.

## Notes

### Acknowledgements

The author would like to acknowledge the support of the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine. This research was funded by the Swedish research council, dnr. 2017-03863.

## References

- AFaF.Alho, A., Fournodavlos, G., Franzen, A.T.: The wave equation near flat Friedmann–Lemaître–Robertson–Walker and Kasner Big Bang singularities
**(preprint)**. arXiv:1805.12558 - ARen.Allen P., Rendall A.D.: Asymptotics of linearized cosmological perturbations. J. Hyperbol. Differ. Equ.
**7**(2), 255–277 (2010)MathSciNetCrossRefGoogle Scholar - AaR.Andersson L., Rendall A.D.: Quiescent cosmological singularities. Commun. Math. Phys.
**218**, 479–511 (2001)ADSMathSciNetCrossRefGoogle Scholar - Bac.Bachelot, A.: Wave asymptotics at a cosmological time-singularity
**(preprint)**. arXiv:1806.01543 - Bre.Brehm, B.: Bianchi VIII and IX vacuum cosmologies: almost every solution forms particle horizons and converges to the Mixmaster attractor
**(preprint)**. arXiv:1606.08058 - CaG.Choquet-Bruhat Y., Geroch R.: Global aspects of the Cauchy problem in general relativity. Commun. Math. Phys.
**14**, 329–335 (1969)ADSMathSciNetCrossRefGoogle Scholar - FaR.Friedrich, H., Rendall, A.D.: The Cauchy problem for the Einstein equations. In: Einstein’s Field Equations and Their Physical Implications. Lecture Notes in Physics, vol. 540, pp. 127–223. Springer, Berlin (2000)CrossRefGoogle Scholar
- HaW.Hewitt C.G., Wainwright J.: A dynamical systems approach to Bianchi cosmologies: orthogonal models of class B. Class. Quantum Grav.
**10**, 99–124 (1993)ADSMathSciNetCrossRefGoogle Scholar - IOaP.Ilchmann A., Owens D.H., Prätzel-Wolters D.: Sufficient conditions for stability of linear time varying systems. Syst. Control Lett.
**9**, 157–163 (1987)MathSciNetCrossRefGoogle Scholar - KraEtal.Krasiński A., Behr C.G., Schücking E., Estabrook F.B., Wahlquist H.D., Ellis G.F.R., Jantzen R., Kundt W.: The Bianchi classification in the Schücking–Behr approach. Gen. Rel. Grav.
**35**(3), 475–489 (2003)ADSCrossRefGoogle Scholar - Lee.Lee J.M.: Introduction to Smooth Manifolds, 2nd edn. Springer, New York (2013)Google Scholar
- LaW.Lin X.-F., Wald R.: Proof of the closed-universe-recollapse conjecture for diagonal Bianchi type-IX cosmologies. Phys. Rev. D
**40**, 3280–3286 (1989)ADSMathSciNetCrossRefGoogle Scholar - Pet.Lindblad Petersen, O.: The mode solution of the wave equation in Kasner spacetimes and redshift. Math. Phys. Anal. Geom. 19, no. 4, Art. 26 (2016)Google Scholar
- Mis.Misner C.W.: Mixmaster universe. Phys. Rev. Lett.
**22**, 1071–1074 (1969)ADSCrossRefGoogle Scholar - RadNon.Radermacher, K.: Strong Cosmic Censorship in orthogonal Bianchi class B perfect fluids and vacuum models
**(preprint)**. arXiv:1612.06278 - RadSti.Radermacher, K.: Orthogonal Bianchi B stiff fluids close to the initial singularity
**(preprint)**. arXiv:1712.02699 - RinAtt.Ringström H.: The Bianchi IX attractor. Annales Henri Poincaré
**2**, 405–500 (2001)ADSMathSciNetCrossRefGoogle Scholar - RinCau.Ringström H: The Cauchy Problem in General Relativity. European Mathematical Society, Zürich (2009)CrossRefGoogle Scholar
- RinSta.Ringström H.: On the Topology and Future Stability of the Universe. Oxford University Press, Oxford (2013)CrossRefGoogle Scholar
- Rin.Ringström, H.: Linear systems of wave equations on cosmological backgrounds with convergent asymptotics
**(preprint)**. arXiv:1707.02803 - RaSLBB.Rodnianski I., Speck J.: A regime of linear stability for the Einstein–Scalar field system with applications to nonlinear big bang formation. Ann. Math.
**187**, 65–156 (2018)MathSciNetCrossRefGoogle Scholar - RaSBB.Rodnianski, I., Speck, J.: Stable big bang formation in near-FLRW solutions to the Einstein–Scalar Field and Einstein–Stiff fluid systems
**(preprint)**. arXiv:1407.6298 - SpeBB.Speck, J.: The maximal development of near-FLRW data for the Einstein-scalar field system with spatial topology \({\mathbb{S}^3}\)
**(preprint)**. arXiv:1709.06477 - WaH.Wainwright J., Hsu L.: A dynamical systems approach to Bianchi cosmologies: orthogonal models of class A. Class. Quantum Grav.
**6**, 1409 (1989)ADSMathSciNetCrossRefGoogle Scholar

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